This space is the universal compact metric space i
This space is the universal compact metric space in the sense that there is a continuous surjection from it to any other non-empty compact metric space. A "wild" example of this space which is embedded in R^3 [read "R three"] with its complement not simply connected is Antoine's necklace. Though it has measure zero, analogues with positive measure that are still nowhere dense can be constructed as so-called "fat" examples. The Devil's staircase function has derivative zero on the complement of this set but is not constant. By the Moore-Kline theorem every non-empty, compact, perfect, and totally disconnected metric space is homeomorphic to it. A multi-dimensional version is called this type of dust. For 10 points, name this set, constructed by starting with the unit interval and then successively removing the middle-thirds of all intervals in sight.
This space is the universal compact metric space in the sense that there is a continuous surjection from it to any other non-empty compact metric space.
{Cantor} set or {Cantor} space or {Cantor} dust